What is the distance between a 20-solar-mass star and neutron star orbiting each other?

If a 20-solar-mass star and a neutron star orbit each other every 13.1 days, then what is the average distance between them?



I know that you need to use Newtons version of Keplers Third Law. M1 would be 20 and I think P=13.1/365?What is the distance between a 20-solar-mass star and neutron star orbiting each other?Newton's version of Kepler's third law tells you that

d[AU]^3 = P[years]^2 * M[SolarMasses]



In this case P = 13.1/365.25 = 0.036 year



M is the mass summed of the two objects. Neutron stars have masses between 1.4 and 2.7 M_Sun, so that is 2 M_Sun with an error of 0.6 M_Sun. So use M=22 M_Sun.

Then d[AU]^3 = 0.036^2 * 22 = 0.0285 and d=0.305 AU.



The error of 0.6 M_Sun is an error of 0.6/22 = 2.7% on M and therefore only an error of 0.9% on d. So we can write: d=0.305 +- 0.003 AUWhat is the distance between a 20-solar-mass star and neutron star orbiting each other?Did they give you the mass of the neutron star? If they did, then you need to use reduced mass to get the period, the formula being:



R = P^(2/3) * (G(M1 + M2)/4%26amp;pi^2)^1/3 .... ref: http://hyperphysics.phy-astr.gsu.edu/hba鈥?/a>



But if you don't know M2, or it is much smaller than M1, than just solve as follows:



F = M1 M2 G/r^2 ... Newton's universal law of gravity

F = M2 v^2/r .... centripetal force



M1 M2 G/r^2 = M2 v^2/r

M1 G/r = v^2



But v = circumference of orbit divided by period.

v = 2蟺 r / P



Subing in

M1 G/r = (2蟺 r/P)^2

P^2 M1 G = 4 蟺^2 r^3



===%26gt; r = P^2/3 * (G M1 /4蟺^2)^1/3

You should note the similarity btw this formula and the one above.



P = 13.1 days = 1.132 x10^6 s ... you convert to seconds because that is the unit "G" is in

M1 = 20 * 2x10^30 kg ... the mass of the sun is 2x10^30 kg

G = 6.67 x10^-11 m^3/kg鈰卻^2



Plugging everything in:



====%26gt; r = 4.424 x10^10 m